Author: natbanting

Constraining the Two-Column Proof

Post author By natbanting
Post date April 13, 2017
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There is no dedicated course for geometry in Saskatchewan’s secondary curriculum. Instead, the topic is splintered amongst several courses. There are advantages and disadvantages to this, neither of which will be the focus of this post. I just thought that, especially for the non-Canadian crowd, a glimpse of context would be helpful.

The notion of a geometric proof only appears in one course. It is presented as a single unit of study during a Grade 11 course and is preceded by a short unit on the difference between inductive and deductive reasoning. I have taught this course a lot over the past few years, and have always had mixed emotions toward this portion. I love the metacognitive analysis students participate in during the inductive v. deductive reasoning unit. It is a (metric) tonne of fun to teach because it largely entails the completion of games, puzzles, or challenges and a subsequent interrogation of our thinking patterns. This could be my favourite week and a half in the course. After we have experienced the difference between induction and deduction, we spend a couple weeks slogging through angle relationships and parallel lines, triangles, and polygons using the ultimate edifice of deductive reason: The two-column proof.

Experiencing Scale in Higher Dimensions

Post author By natbanting
Post date March 25, 2017
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A colleague and I have often bemoaned our attempts to teach the concept of scale factor in higher dimensions. A topic that has such beautiful and elegant patterns and symmetries between the scale factors consistently seems to sail directly past the experience of our students. I have tried enacting several tasks with the students including some favourites from the #MTBoS (Mathalicious 1600 Pennsylvania and Giant Gummy Bear). Each time, the thinking during the task seems to dissipate when new problems are offered. It just seems like students have a hard time trusting the immense rate that surface area and volume can grow (or shrink). In the past, I had used digital images of cubes growing after having their dimensions scaled by 2, 3, 4… etc.; students seemed to grasp the pattern yet under-appreciate the girth of 8, 27, 64… etc. times as many cubes.

area cylinder surface area tasks volume

Solid Fusing Task

The progression followed by most teachers and resources during the study of surface area and volume is identical. Like a intravenous drip, concepts are released gradually to the patients so as to not overdose them with complexity. Begin with the calculation of 2-dimensional areas, and then proceed to the calculation of surface area of familiar prisms. (I say prisms, so a parallel can be drawn to the common structure for finding the volume of said prisms. That is, [area of base x height]). In this way, surface area is conceptualized as nothing more than a dissection of 3-dimensional solids into the now familiar 2-dimensional shapes.

reflection

Real-World: An Attack on “Relevance”

Post author By natbanting
Post date November 1, 2016
6 Comments on Real-World: An Attack on “Relevance”

**deep breath**

Last week, I caught myself saying something to a pre-service teacher as we planned a Grade 1 lesson for the making of 10s. I asked her,

“Why would the students need to know how to make up 10s?”

When she was auspiciously silent, I filled the space with a statement said entirely tongue-in-cheek. It was only upon reflection, that I kicked myself for not being able to shut up and allow her to think. I said,

“…because my job is to convince teenagers they need logarithms, and that is much more difficult.”

factors probability tasks

100 Rolls Task

**Update November 2020. Jamie Mitchell–a fantastic teacher from Ontario, Canada–sent me this google doc that his student prepared to justify her solution. After you wrestle with the prompt for a while, take a second to read this brilliant response!

Most probability resources contain a familiar type of question: the two-dice probability distribution problem.

Often times, it is accompanied with questions concerning the sums of the faces that appear on each dice.

For example:

Roll two fair, 6-sided dice. What possible sums can be made by adding the faces together?

What is the probability that:

a) the sum is 6

b) the sum is a multiple of 4

c) the sum is greater than 15?

equations estimation games pattern reflection stations tasks

TDC Math Fair 2016: A Summary

Post author By natbanting
Post date June 29, 2016
3 Comments on TDC Math Fair 2016: A Summary

Background:
On June 15th, my Grade 9 class and I hosted our second annual math fair. What started out as a small idea has grown into a capstone event of their semester. This year, we had 330 elementary school students visit our building to take part in the fair’s activities. Several people (following the hashtag #TDCMathFair2016) commented that they would like to do similar things with their student transitions. This post details the rationale behind the event, how we structured it, what stations we had, and feedback/advice from our exploits.

reflection

Mathematics Is: Student Impressions

Post author By natbanting
Post date June 20, 2016
4 Comments on Mathematics Is: Student Impressions

I have taught the second half of a Math 9 Enriched course for the last three years. The students generally finish two-thirds of the curricular outcomes during the first semester (with an different teacher). This alleviates the perpetual nemesis of time, and leaves me with no excuse to stretch the boundaries of what is possible in a classroom.

I spend most of the time developing a classroom ecology focused around conjecture, community, and curiosity. The result is a constant focus on problem shaping, solving, and re-posing.

At the end of the semester, I ask students to respond to a simple prompt. They have ten minutes to answer:

What is Mathematics?

numeracy systems of equations tasks

MVPs and Fair Teams

You will not catch me claiming that problems need to be real world in order to be relevant. I would much rather think of classroom materials as either mind numbing or thought provoking. This continuum can be applied to hypothetical, practical, or genuine scenarios (a classification system neatly summarized in a chart in this article).

I see the greatest potential in scenarios that provide elegant entrance to mathematical reasoning. If it happens to contain a real world context, fantastic. Either way, it needs to be thought provoking.

games numeracy order of operations

(Min + Max) imize: A Classroom Game for Basic Facts

Post author By natbanting
Post date May 8, 2016
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**this post was elaborated on in the May 2016 issue of The Variable from the SMTS.

**this game is part of a larger collection of classroom games entitled, 16 Boxes: Ten Numeracy Routines for the Classroom available (for free) at this link.

This is a game that was adapted from a colleague in my department. He can’t quite remember where it came from, but knows there was some influence from his undergraduate days. Nonetheless, he reinvented it to play with his Grade 9s, and this post represents yet another reinvention.

The game has a simple mechanism (dice rolling), and endless extensions to elaborate on and play with. These are both keys to a great classroom game (for me anyway).

(Min + Max) imize practices basic operations within the framework of larger, conceptual decision making. While I rarely bring up the probability of the dice rolling, it is obvious that students are making decisions based on the chances of certain rolls being obtained. The idea is to practice basic skills and order of operations in a way that allows students to be active, numerate decision makers.

games integers

Limbo: An Integers Game

Rationale: Create a game that embeds the skills of adding and subtracting integers into a conceptual decision making structure.

Objective: Insert a set of integers into a 4-by-4 grid so that the sums of the rows and columns is a minimum.

Game Set-up:
All the students need is the game board and the list of sixteen numbers.